Hostname: page-component-cd4964975-8tfrx Total loading time: 0 Render date: 2023-03-31T14:06:24.640Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true

Looking for Those Natural Numbers: Dimensionless Constants and the Idea of Natural Measurement1

Published online by Cambridge University Press:  26 September 2008

Philip Mirowski
Department of Economics and Reilly Center for the History and Philosophy of ScienceUniversity of Notre Dame


Many find it “notoriously difficult to see how societal context can affect in any essential way how someone solves a mathematical problem or makes a measurement.” That may be because it has been a habit of western scientists to assert their numerical schemes were untainted by any hint of anthropomorphism. Nevertheless, that Platonist penchant has always encountered obstacles in practice, primarily because the stability of any applied numerical scheme requires some alien or external warrant.

This paper surveys the history of measurement standards, physical dimensions and dimensionless constants as one instance of the quest to purge all anthropomorphic taint first in the metric system, then in the dimensions provided by the atom, then in physical constants intelligible to extraterrestrials, only then to end up back at overt anthropomorphism in the late 20th century. This suggests that the “naturalness” of natural numbers has always been conceptualized in locally contingent cultural terms.

Research Article
Copyright © Cambridge University Press 1992

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



This paper was inspired by some remarks by Martin Klein during his lectures at Yale on the history of early twentieth-century physics; but he cannot be held responsible for my interpretations. I also thank Ted Porter for his help with the philosophical issues.


Barrow, J., and Tippler., F. 1986. The Anthropic Cosmological Principle. Oxford: Oxford University Press.Google Scholar
Beller, Mara. 1988. “Experimental Accuracy, Operationalism, and the Limits of Knowledge.” Science in Context 2:147–62.CrossRefGoogle Scholar
Bloor, David. 1982. “Durkheim and Mauss Revisited: Classification and the Sociology of Knowledge.” Studies in the History and Philosophy of Science 13:267–97.CrossRefGoogle Scholar
Bridgman, Percy. 1931. Dimensional Analysis, 2nd. ed. New Haven, Conn..: Yale University Press.Google Scholar
Buckingham, E. 1914. “On Physically Similar Systems.” Physical Reviews 4:347–76.CrossRefGoogle Scholar
Cahan, David. 1989. An Institute for an Empire. New York: Cambridge University Press.Google Scholar
Campbell, Norman. 1924. “Ultimate Rational Units.” Philosophical Magazine 47:159–72.Google Scholar
Campbell, Norman. [1920] 1957. Foundations of Science. New York: Dover.Google Scholar
Carr, B., and Rees., M. 1979. “The Anthropic Principle and the Structure of the Physical World.” Nature 278:605–12.CrossRefGoogle Scholar
Causey, Robert. 1969. “Derived Measurement, Dimensions, and Dimensional Analysis.” Philosophy of Science 36:252–70.CrossRefGoogle Scholar
Chandrasekhar, S. 1937. “The Cosmological Constants.” Nature 139:757–78.CrossRefGoogle Scholar
Cochrane, Rexmond. 1966. Measures for Progress. Washington, D.C.: National Bureau of Standards.Google Scholar
Cohen, E., Crowe, K., and Dumond., J. 1957. The Fundamental Constants of Physics. New York: Interscience Publishers.Google Scholar
De Finetti, Bruno. 1974. Theory of Probability. New York: Wiley.Google Scholar
Deutsch, David. 1986. “On Wheelers' Notion of Law without Law in Physics.” Foundations of Physics 16:565–73.CrossRefGoogle Scholar
Dirac, Paul. 1937. “The Cosmological Constants.” Nature 139:323.CrossRefGoogle Scholar
Douglas, Mary. 1986. How Institutions Think. Syracuse, N.Y.: Syracuse University Press.Google Scholar
Durbin, John. 1985. Modern Algebra, 2nd ed. New York: Wiley.Google Scholar
Eddington, Arthur. 1930. The Nature of the Physical World. London: Macmillan.Google Scholar
Eddington, Arthur. 1935. New Pathways in Science. Cambridge: Cambridge University Press.Google Scholar
Eddington, Arthur. 1936. Relativity Theory of Protons and Electrons. Cambridge: Cambridge University Press.Google Scholar
Eddington, Arthur. 1939. Philosophy of Physical Science. Cambridge: Cambridge University Press.Google Scholar
Eddington, Arthur. 1941. “Group Structure in Physical Science.” Mind 50:268–79.CrossRefGoogle Scholar
Einstein, Albert. 1911. “Elementare Betrachtungen über die thermische Molekularbewegungen in festen Körpern.” Annalen der Physik 35:679–94.Google Scholar
Fourier, Joseph. 1878. The Analytical Theory of Heat. Cambridge: Cambridge University Press.Google Scholar
Frieden, B. R. 1986. “A Probability Law for Fundamental Constants.” Foundations of Physics 16:883903.CrossRefGoogle Scholar
Galison, Peter. 1987. How Experiments End. Chicago: University of Chicago Press.Google Scholar
Guggenheim, E. 1942. “Units and Dimensions.” Philosophical Magazine 33:479–96.Google Scholar
Hacking, Ian. 1983. Representing and Intervening. New York: Cambridge University Press.CrossRefGoogle Scholar
Halsey, Frederick. 1920. The Metric Fallacy, 2nd ed. New York: American Institute of Weights and Measures.Google Scholar
Heilbron, J. L. 1986. Dilemmas of an Upright Man. Berkeley: University of California Press.Google Scholar
Heilbron, J. L. 1990. “The Measure of Enlightenment.” In The Quantifying Spirit in the Eighteenth Century, edited by Frängsmyr, Tore et al. Berkeley: University of California Press.Google Scholar
Heilig, Klaus. 1978. “The Dutch Book Argument Reconsidered.” British Journal for the Philosophy of Science 29:325–46.CrossRefGoogle Scholar
Jevons, W. S. [1874] 1905. The Principles of Science, 3rd ed. London: Macmillan.Google Scholar
Jevons, W. S. [1871] 1970. The Theory of Political Economy. Baltimore: Penguin.Google Scholar
Jordan, Pascual. 1939. Annalen der Physik 36:64.CrossRefGoogle Scholar
Jordan, Pascual. 1949. “The Formation of the Stars and the Development of the Universe.” Nature 164:637–40.CrossRefGoogle Scholar
Klein, Martin. 1977. “The Beginnings of Quantum Theory.” In History of Twentieth Century Physics, edited by Weiner., C. New York: Academic Press.Google Scholar
Kothari, D. 1938. “Cosmological and Atomic Constants.” Nature 142:354–55.CrossRefGoogle Scholar
Krantz, D., Luce, R., Suppes, P., and Tversky., A. 1971. Foundations of Measurement, Vol. 1. New York: Academic Press.Google Scholar
Kula, Witold. 1986. Measures and Men. Princeton, N.J.: Princeton University Press.CrossRefGoogle Scholar
Kverneland, K. 1978. World Metric Standards for Engineering. New York: Industrial Press.Google Scholar
Lewis, Gilberg. 1925. “Ultimate Rational Units and Dimensional Theory.” Philosophical Magazine 9:739–50.Google Scholar
Lundgreen, Peter. 1986. Standardization — Testing — Regulation. Bielefeld: Kleine Verlag.Google Scholar
Macleod, H. 1884. “Address to the Board of Electors to the Professorship of Political Economy in the University of Cambridge.” London.Google Scholar
Manin, Yu. 1981. Mathematics and Physics. Boston: Birkhauser.CrossRefGoogle Scholar
Maxwell, James Clerk. 1870. “Presidential Address to Section A of the British Association.” Nature 2:419–22.Google Scholar
Maxwell, James Clerk. 1871. The Theory of Heat. London: Longmans.Google Scholar
Maxwell, James Clerk. 1965. The Scientific papers. Edited by Niven., W. New York: Dover.Google Scholar
Meyerson, Emile. [1908] 1962. Identity and Reality. New York: Dover.Google Scholar
Miller, Arthur. 1986. Imagery in Scientific Thought. Cambridge, Mass.: MIT Press.Google Scholar
Mirowski, Philip. 1989. More Heat than Light. New York: Cambridge University Press.CrossRefGoogle Scholar
Mirowski, Philip. 1991. “Postmodernism and the Social Theory of Value.” Journal of Post Keynesian Economics. 13:565–82.CrossRefGoogle Scholar
Moore, Walter. 1989. Schrödinger: Life and Thought. New York: Cambridge University Press.CrossRefGoogle Scholar
Norman, Eric. 1986. “Are the Fundamental Constants Really Constant?American Journal of Physics 54:317–21.CrossRefGoogle Scholar
Ohanian, Hans. 1977. “Cosmological Changes in Atomic and Nuclear Constants.’ Foundations of Physics 7:391403.CrossRefGoogle Scholar
Osborne, D. 1978. “On Dimensional Invariance.” Quantity and Quality 12:7589.CrossRefGoogle Scholar
Pais, Abraham. 1982. Subtle Is the Lord … Oxford: Oxford University Press.Google Scholar
Petley, B. W. 1985. The Fundamental Physical Constants and the Frontiers of Measurement. Bristol: Adam Hilger.Google Scholar
Planck, Max. 1931. The Universe in Light of Modern Physics. New York: Norton.Google Scholar
Planck, Max. 1949. Scientific Autobiography. New York: Philosophical Library.Google Scholar
Planck, Max. 1959. The Theory of Heat Radiation. New York: Dover.Google Scholar
Pribram, Karl. 1983. A History of Economic Reasoning. Baltimore; Johns Hopkins University Press.Google Scholar
Rankine, W. 1881. Miscellaneous Scientific Papers. London: Charles Griffin.Google Scholar
Ringer, Fritz. 1969. The Decline of the German Mandarins. Cambridge, Mass.: Harvard University Press.Google Scholar
Rosenthal-Schneider, Ilse. 1980. Reality and Scientific Truth. Detroit: Wayne State University Press.Google Scholar
Rucker, A. 1889. “On the Suppressed Dimensions of Physical Quantities.” Philosophical Magazine 27:104–14.Google Scholar
Ruger, Alexander. 1988. “Atomism from Cosmology.” Historical Studies in the Physical Sciences 18:376401.Google Scholar
Shapin, S., and Schaffer., S. 1985. Leviathan and the Air Pump. Princeton, N. J.: Princeton University Press.Google Scholar
Stoney, G. Johnstone 1881. “On the Physical Units of Nature.” Philosophical Magazine 11:381–90.Google Scholar
Tiles, Mary. 1984. Bachelard: Science and Objectivity. New York: Cambridge University Press.CrossRefGoogle Scholar
Whittaker, Edmund. 1945. “Eddington's Theory of the Constants of Nature.” Mathematical Gazette 29:137–44.Google Scholar
Whittaker, Edmund. 1951. Eddington's Principle in the Philosophy of Science. Cambridge: Cambridge University Press.Google Scholar
Wise, M. N. 1988. “Mediating Machines.” Science in Context 2(1):77114.CrossRefGoogle Scholar